CHAPTER 16 Broadcast Scheduling for TDMA in Wireless Multihop Networks ERROL L. LLOYD Department of Computer Science and Information Sciences, University of Delaware 16.1 INTRODUCTION Wireless multihop networks, also known as packet radio networks and ad hoc networks, hold great promise for providing easy to use mobile services for many applications, espe- cially military and disaster relief communications. Such networks can provide robust communication, be rapidly deployed, and respond quickly in dynamic environments. However, effectively deploying and utilizing such networks poses many technical chal- lenges. One such challenge is to make effective use of the limited channel bandwidth. In this chapter, we describe one approach to this challenge, namely broadcast scheduling of channel usage by way of TDMA (time division multiple access). Our emphasis is on the fundamental computational and algorithmic issues and results associated with broadcast scheduling. The chapter is organized as follows. In the next section we provide background and ter- minology on broadcast scheduling and related topics. Section 16.3 examines the computa- tional complexity of broadcast scheduling. Sections 16.4 and 16.5 study approximation al- gorithms in centralized and distributed domains. Section 16.6 briefly outlines some related results. Finally, Section 16.7 summarizes the chapter and outlines prominent open problems. 16.2 WHAT IS BROADCAST SCHEDULING? Background and terminology associated with wireless multihop networks, the modeling of such networks, and related concepts are provided in this section. 16.2.1 Basic Concepts of Wireless Multihop Networks We define a wireless multihop network as a network of stations that communicate with each other via wireless links using radio signals. All of the stations share a common chan- nel. Each station in the network acts both as a host and as a switching unit. It is required 347 Handbook of Wireless Networks and Mobile Computing, Edited by Ivan Stojmenovic´ Copyright © 2002 John Wiley & Sons, Inc. ISBNs: 0-471-41902-8 (Paper); 0-471-22456-1 (Electronic) that the transmission of a station be received collision-free by all of its one-hop (i.e., di- rect) neighbors. This cannot occur if a station transmits and receives simultaneously or if a station simultaneously receives from more than one station. A collision caused by trans- mitting and receiving at the same time is called a primary conflict. A collision caused by simultaneously receiving from two stations is called a secondary conflict. We note that as a practical matter, some existing multihop networks may violate one or more of the above assumptions. A wireless multihop network can be modeled by a directed graph G = (V, A ), where V is a set of nodes denoting stations in the network and A is a set of directed edges between nodes, such that for any two distinct nodes u and v, edge (u, v) ʦ A if and only if v can re- ceive u’s transmission. As is common throughout the literature, we assume that (u, v) ʦ A if and only if (v, u) ʦ A. That is, links are bidirectional, in which case it is common to use an undirected graph G = (V, E). Throughout this chapter we use undirected graphs to model wireless multihop networks. Sharing a common channel introduces the question of how the channel is accessed. Channel access mechanisms for wireless multihop networks fall into two general cate- gories: random access (e.g., ALOHA) and fixed access. Broadcast scheduling, the focus of this chapter, is a fixed access technique that preallocates the common channel by way of TDMA so that collisions do not occur. 16.2.2 Defining Broadcast Scheduling The task of a broadcast scheduling algorithm is to produce and/or maintain an infinite schedule of TDMA slots such that each station is periodically assigned a slot for transmis- sion and all transmissions are received collision-free. In this framework, most broadcast scheduling algorithms operate by producing a finite length nominal schedule in which each station is assigned exactly one slot for transmission, and then indefinitely repeating that nominal schedule. Except where noted otherwise, throughout this chapter the term broadcast schedule refers to a nominal schedule. 16.2.3 Graph Concepts and Terminology In modeling wireless multihop networks by undirected graphs, many variants are possible in regard to network topology. Among the possibilities, the three most relevant to this chapter are: 1. Arbitrary graphs. Such graphs can model any physical situation, including for ex- ample, geographically close neighbors that cannot communicate directly due to in- terference (e.g., a mountain) on a direct line between the stations. 2. Planar graphs. A graph is planar if and only if it can be drawn in the plane such that no two edges intersect except at common endpoints. Planar graphs are among the most widely studied classes of graphs. 3. Unit disk graphs. Formally introduced in [9] for use in network modeling, and stud- ied in conjunction with broadcast scheduling in [28], unit disk graphs [9, 3] model 348 BROADCAST SCHEDULING FOR TDMA IN WIRELESS MULTIHOP NETWORKS the situation in which all stations utilize a uniform transmission range R, and there is no interference. Thus, the transmission of a station v will be received by all sta- tions within a Euclidean distance R of v. In these graphs, there is an edge between nodes u and v if and only if the Euclidean distance between stations u and v does not exceed R. An example of a unit disk graph model of a wireless multihop net- work and a nominal schedule are shown in Figure 16.1. In that figure, there is a link between a pair of stations if and only if the circles of radius R/2 centered at the pair of stations intersect, including being tangent. The slots assigned to the stations are the numbers inside the brackets. Regardless of the graph model utilized, if there is an edge between nodes u and v, then u is a one-hop neighbor/neighbor of v and likewise v is a neighbor of u. The degree of a node is the number of neighbors. The degree ␳ of a network is the maximum degree of the nodes in the network. The distance-2 neighbors of a node include all of its one-hop neigh- bors and the one-hop neighbors of its one-hop neighbors. The two-hop neighbors of a node are those nodes that are distance-2 neighbors, but are not one-hop neighbors. The unit subset of node u consists of u and its distance-2 neighbors. The distance-2 degree D(u) of u is the number of distance-2 neighbors of u, and the distance-2 degree D of a net- work, is the maximum distance-2 degree of the nodes in the network. Relevant to broadcast scheduling is distance-2 coloring [22, 13] of a graph G = (V, E), where the problem is to produce an assignment of colors C : V Ǟ 1, 2, . . . such that no two nodes are assigned the same color if they are distance-2 neighbors. An optimal color- ing is a coloring utilizing a minimum number of colors. A distance-2 coloring algorithm is said to color nodes in a greedy fashion (i.e., greedily) if when coloring a node, the color 16.2 WHAT IS BROADCAST SCHEDULING? 349 Figure 16.1 Broadcast scheduling modeled by a unit disk graph. assigned is the smallest number color that can be assigned without resulting in conflicts. Here, the constraint number of a node is the number of different colors assigned to the node’s distance-2 neighbors. In the context of broadcast scheduling, determining a nominal schedule is directly ab- stracted to distance-2 coloring, whereby slots that are assigned to stations are translated into colors that are assigned to nodes. In this chapter, we will interchangeably use the terms network and graph, station and node, and slot and color. 16.2.4 Varieties of Broadcast Scheduling Algorithms There are two main varieties of broadcast scheduling algorithms—centralized and distrib- uted. Centralized Algorithms Centralized algorithms are executed at a central site and the results are then transmitted to the other stations in the network. This requires that the central site have complete informa- tion about the network. This is a strong assumption that is not easy to justify for wireless multihop networks with mobile stations. The study of centralized algorithms, however, provides an excellent starting point for both the theory of broadcast scheduling and the de- velopment of more practical algorithms. Further, for some stationary wireless networks, it is reasonable to run a centralized algorithm at the net management center and then distrib- ute schedules to stations. In the centralized algorithm context, there are two types of algorithms corresponding to how the input is provided: 1. Off-line algorithms. The network topology is provided to the central site in its en- tirety. The algorithm computes the schedule for the entire network once and for all. 2. Adaptive algorithms: With off-line algorithms, if the network topology changes, then the algorithm is rerun for the entire network. However, as wireless networks are evolving towards thousands of stations spread over a broad geographical area and operating in an unpredictable dynamic environment, the use of off-line schedul- ing algorithms is not realistic. In practice, it is absolutely unaffordable to halt com- munication whenever there is a change in the network, so as to produce a new schedule “from scratch.” In such circumstances, adaptive algorithms are required That is, given a broadcast schedule for the network, if the network changes (by the joining or leaving of a station), then the schedule should be appropriately updated to correspond to the modified network. Thus, an adaptive algorithm for broadcast scheduling is one that, given a wireless multihop network, a broadcast schedule for that network, and a change in the network (i.e., either a station joining or leaving the network), produces a broadcast schedule for the new network. The twin objectives of adaptive algorithms are much faster execution (than an off-line algorithm that computes a completely new schedule) and the production of a provably high-quality schedule. We note that many other network changes can be modeled by joining or leaving or a combination of the two (e.g., the moving of a station from one location to another). 350 BROADCAST SCHEDULING FOR TDMA IN WIRELESS MULTIHOP NETWORKS Distributed Algorithms Although centralized algorithms provide an excellent foundation, algorithms in which the computation is distributed among the nodes of the network are essential for use in prac- tice. In these distributed algorithms, network nodes have only local information and par- ticipate in the computation by exchanging messages. Distributed algorithms are important in order to respond quickly to changes in network topology. Further, the decentralization results in decreased vulnerability to node failures. We distinguish between two kinds of distributed algorithms: 1. Token passing algorithms. A token is passed around the network. When a station holds the token, it computes its portion of the algorithm [26, 2]. There is no central site, although a limited amount of global information about the network may be passed with the token. Token passing algorithms, while distributing the computa- tion, execute the algorithm in an essentially sequential fashion. 2. Fully distributed algorithms. No global information is required (other than the glob- al slot synchronization associated with TDMA), either in individual or central sites. Rather, a station executes the algorithm itself after collecting information from sta- tions in its local vicinity. Multiple stations can simultaneously run the algorithm, as long as they are not geographically too close, and stations in nonlocal portions of the network can transmit normally even while other stations are joining or leaving the network. Fully distributed algorithms are essentially parallel, and are typically able to scale as the network expands. 16.3 THE COMPLEXITY OF BROADCAST SCHEDULING In this section the computational complexity of broadcast scheduling is studied. 16.3.1 Computing Optimal Schedules As noted in the previous section, determining a minimum nominal schedule in a wireless multihop network is equivalent to finding a distance-2 graph coloring that uses a mini- mum number of colors. The NP-completeness of distance-2 graph coloring is well estab- lished [22, 6, 26, 5]. The strongest of these [6] shows that distance-2 graph coloring re- mains NP-complete even if the question is whether or not four colors will suffice. They utilize a reduction from standard graph coloring. Thus: Theorem 1 Given an arbitrary graph and an integer k Ն 4, determining if there exists a broadcast schedule of length not exceeding k, is NP-complete. By way of contrast, in [25] it is shown that when k is three, the problem can be solved in polynomial time. Given the NP-completeness of the basic problem, we are left with the possible approaches of utilizing approximation algorithms to determine approximately minimal solutions, or considering the complexity on restricted classes of graphs. Most of the remainder of this chapter is devoted to the former. In regard to the latter, we note: 16.3 THE COMPLEXITY OF BROADCAST SCHEDULING 351 Theorem 2 [25] Given a planar graph, determining if there exists a broadcast schedule of length not exceeding seven, is NP-complete. 16.3.2 What About Approximations? From the NP-completeness results cited above, finding minimum length broadcast schedules is generally not possible. Thus, it is necessarily the case that we focus on al- gorithms that produce schedules that are approximately minimal. For such an approxi- mation algorithm, its approximation ratio ␣ [8, 10], is the worst case ratio of the length of a nominal schedule produced by the algorithm to the length of an optimal nominal schedule. Such an algorithm is said to produce ␣ -approximate solutions. In the context of adaptive algorithms, the analagous concept is that of a competitive ratio. Here, the ra- tio is the length of the current nominal schedule produced by the algorithm to an opti- mal off-line nominal schedule. Additional information on such ratios and alternatives may be found in [8, 10]. What quality of approximation ratio might be possible for broadcast scheduling? Most often, the goal in designing approximation algorithms is to seek an approximation ratio that does not exceed a fixed constant (i.e., a constant ratio approximation algorithm). One would hope, as in bin packing and geometric traveling salesperson [10], that approxima- tion ratios of two or less might be possible. Unfortunately, this is not the case, not only for any fixed constant, but also for much larger ratios: Theorem 3 [1] Unless NP = ZPP, broadcast scheduling of arbitrary graphs cannot be approximated to within O(n 1/2– ⑀ ) for any ⑀ > 0. This result is tight since there is an algorithm (see the next section) having an approxi- mation ratio that is O(n 1/2 ). 16.4 CENTRALIZED ALGORITHMS Since broadcast scheduling is NP-complete, in this section (and the next) we investigate approximation algorithms that are alternatives to producing optimal schedules. These al- gorithms are evaluated on the basis of their running times and approximation ratios. 16.4.1 A Classification of Approximation Algorithms for Broadcast Scheduling An overview of approximation algorithms for broadcast scheduling is given in this sec- tion. Only a few particular algorithms are specifically described, and the reader is referred to [11, 25] for a more comprehensive treatment. In developing approximation methods for broadcast scheduling, the classic algorithm P_Greedy takes a purely greedy approach. That algorithm is an iterative method in which a node is arbitrarily chosen from the as yet uncolored nodes and is greedily colored. The running time of P_Greedy is O(n ␳ 2 ) on arbitrary graphs and O(n ␳ ) on unit disk graphs. 352 BROADCAST SCHEDULING FOR TDMA IN WIRELESS MULTIHOP NETWORKS The approximation ratio of P_Greedy is min( ␳ , n 1/2 ) [26, 22] on arbitrary graphs, and 13 on unit disk graphs [18]. Aside from P_Greedy, a variety of centralized approximation algorithms have been proposed for broadcast scheduling. These algorithms can be placed into three general cat- egories, using a classification adapted from [11]: 1. Traditional algorithms that preorder the nodes according to a specified criterion, and then color the nodes in a greedy fashion according to that ordering. A represen- tative of such methods is Static_min_deg_last [11]. In this method, the nodes are placed into descending order according to their degrees. The nodes are then greedi- ly colored according to that ordering. The running time is O(n min(n, ␳ 2 )) on arbi- trary graphs and O(n log n + n ␳ ) on unit disk graphs. The algorithm is ␳ -approxi- mate on arbitrary graphs, and 13-approximate on unit disk graphs [18]. 2. Geometric algorithms that involve projections of the network onto simpler geomet- ric objects, such as the line. A representative of such methods is Linear_Projection [11], in which the positions of the nodes are projected onto a line, and then an opti- mal distance-2 coloring is computed for those projected points. One effect of pro- jecting nodes onto a line is that the projections of nodes may now be within dis- tance-2, whereas the original nodes were not within distance-2. The algorithm selects a line for projection that minimizes the number of such “false” distance-2 neighbors. Linear_Projection runs in time O(n 2 ) on arbitrary graphs. There are no results on the approximation ratio. 3. Dynamic greedy methods that also color nodes in a greedy fashion, but in which the order of the coloring is determined dynamically as the coloring proceeds. A repre- sentative of such methods is max_cont_color [16]. This algorithm initially colors an arbitrary node and then all of the one-hop neighbors of that node. At each subse- quent step, the algorithm chooses for coloring a node that is now most constrained by its distance-2 neighbors. Results [16, 18] show that this algorithm has the best simulation performance among all existing broadcast scheduling algorithms. A careful implementation [18] yields a running time of O(nD) on arbitrary graphs and O(n ␳ ) on unit disk graphs. The algorithm is ␳ -approximate on arbitrary graphs, and 13-approximate on unit disk graphs [18]. 16.4.2 A Better Approximation Ratio The best approximation ratio for arbitrary graphs of the methods cited above is min(n 1/2 , ␳ ), which is also the ratio of the simplest of these algorithms, P_Greedy. Below, an algorithm of the “traditional greedy” variety is described that has an arguably stronger ratio for most graphs. The algorithm is similar to Static_min_deg_last but the nodes are ordered in a “dynamic,” rather than static, fashion. The term progressive is taken from [23]. Algorithm progressive_min_deg_last(G) Labeler(G, n); for j ǟ 1 to ndo 16.4 CENTRALIZED ALGORITHMS 353 let u be such that L(u) = j; greedily color node u; The function Labeler, which assigns a label between 1 and n to each node, is defined as follows: Labeler(G, ᐉ) if G is not empty let u be a vertex of G of minimum degree; L(u) ǟ ᐉ; Labeler(G – u, ᐉ – 1); Here, G-u is the graph obtained from G by removing u and all incident edges. It is straightforward to see that progressive_min_deg_last produces a legal coloring, since the coloring is performed in a greedy fashion. This will be true for all of the algorithms de- scribed in this chapter, and we will make no further reference to algorithm correctness. Theorem 4 For a planar graph, progressive_min_deg_last is 9-approximate. Proof: Consider the neighbors of an arbitrary node u, and suppose that k of those nodes have labels smaller than L(u). Hence, there are up to ␳ – k neighbors of u with labels larg- er than L(u). It follows from the properties of planar graphs, and the specification of Labeler, that k Յ 5. Each node with a label smaller than L(u) may have at most ␳ – 1 neighbors (not in- cluding u) and hence those k nodes and their neighbors utilize at most k( ␳ – 1) + k = k ␳ colors that may not be assigned to u. Now consider the up to ␳ – k nodes with labels larger than L(u). When u is colored, none of these nodes are colored (recall that coloring is done in increasing order of labels). However, the neighbors of these nodes may have lower labels than L(u) and the already as- signed colors of these nodes may not be assigned to u. Since the minimum node degree in a planar graph is always five or less, it follows from the specification of Labeler that there can be at most 4 · ( ␳ – k) 2-hop neighbors of u that are already colored (i.e., four for each uncolored 1 – hop neighbor of u, not counting u itself). Thus, u can be colored using no more than k ␳ + 4·( ␳ – k) + 1 colors, and with k Յ 5, this is at most 9 ␳ – 19. Since the minimum coloring uses at least ␳ + 1 colors, the approx- imation ratio is bounded above by nine. Ǣ For arbitrary graphs, the approximation ratio of progressive_min_deg_last depends on the thickness of the graph. That is, the minimum number of planar graphs into which the graph may be partitioned. Note that the algorithm does not compute the thickness (indeed, computing the thickness is NP-complete [21]), but rather only the bound depends on that value. Further, experimental results [25] establish that the thickness is generally much less than ␳ . Corollary 1 For an arbitrary graph of thickness ␪ , progressive_min_deg_last has an ap- proximation ratio that is O( ␪ ). 354 BROADCAST SCHEDULING FOR TDMA IN WIRELESS MULTIHOP NETWORKS The corollary follows from the prior proof by noting that in a graph of thickness ␪ , there is at least one node of degree not exceeding 6 ␪ – 1 [25]. The analysis in [14] establishes: Theorem 5 For q-inductive graphs, progressive_min_deg_last has an approximation ra- tio of 2q – 1. Several classes of graphs, including graphs of bounded genus, are q-inductive. See [12] for additional information on q-inductive graphs. In regard to running times, it is shown in [24]: Theorem 6 For planar graphs, progressive_min_deg_last has a running time of O(n ␳ ). For arbitrary graphs of thickness ␪ , progressive_min_deg_last has a running time of O(n ␪␳ ). 16.4.3 A Better Ratio for Unit Disk Graphs Among the methods cited above, several are 13-approximate when applied to unit disk graphs. These ratios follow from a general result of [18] on the performance of “greedy” algorithms. The best approximation ratio relative to unit disk graphs belongs to the follow- ing algorithm (of the traditional greedy variety): Algorithm Continuous_color(G): Let u be an arbitrary node of G; L ǟ a list of the nodes of G sorted in increasing order of Euclidean distance to u; while L  0/ do Let v be the first node of L; Greedily color v and remove v from L; Theorem 7 For unit disk graphs, Continuous_color has an approximation ratio of seven. Key to the proof of this theorem is the following result that establishes that certain nodes in geographical proximity to one another must be distance-2 neighbors. Lemma 1 Given a node p, let b 1 and b 2 be two points on the boundary of p’s interference region (i.e., the circle of radius 2R centered at p). If |b 1 b 2 | Յ R, then any two nodes that are both: ț in the unit subset of p, and ț within the area bounded by the line segments from p to b 1 and b 2 and by the bound- ary of the interference region of p that runs between b 1 and b 2 (we refer to this area as section pb 1 b 2 and show it as the shaded area in Figure 16.2) are distance-2 neighbors. 16.4 CENTRALIZED ALGORITHMS 355 The proof of Lemma 1 involves the extensive use of trigonometric functions to establish the proximity of points, and may be found in [18]. Proof: From Continuous_color, when a node v is to be colored, the already colored dis- tance-2 neighbors of v lie on at most half of v’s interference region. The perimeter of that half of an interference region can be partitioned into seven sections such that the distance between the extreme perimeter points in each section does not exceed R. Within each of these sections, from Lemma 1, all of the nodes are distance-2 neighbors, hence each of the nodes must receive a distinct color. Let x be the largest number of nodes in any one of these sections. Then x + 1 is a lower bound on the number of different colors assigned to v and its already colored neighbors. Likewise, 7x + 1 is an upper bound on the number of different colors used by Continuous_color. The theorem follows from the ratio of the up- per to lower bound. Ǣ In regard to running time, it is easy to see that the running time of Continuous_color is O(n log n +nD) when applied to arbitrary graphs. Since D may be as large as ␳ 2 , we have: Lemma 2 For arbitrary graphs, the running time of Continuous_color is O(n log n + n ␳ 2 ). For unit disk graphs the running time is less. Key to that result is the following, which establishes that D and ␳ are linearly related: Lemma 3 In unit disk graphs, D Յ 25 ␳ . Proof: Consider the interference region of an arbitrary node s in a unit disk graph. Clear- ly, all distance-2 neighbors of s lie within the interference region of s. Now, define an S- cycle to be a circle of radius R/2 (it could be centered anywhere) and note that all nodes lying within any given S-cycle are one-hop neighbors. Thus, at most ␳ nodes lie within any given S-cycle. The lemma follows since the interference region of any node can be cov- ered with 25 S-cycles as shown in Figure 16.3 (in that figure, S-cycles are shown both shaded and unshaded to make them easier to visualize). Ǣ 356 BROADCAST SCHEDULING FOR TDMA IN WIRELESS MULTIHOP NETWORKS Figure 16.2 The locations of b 1 and b 2 . [...]... the O(␳2) time required by Algorithm_IR is much superior for unit disk graphs 16.5 DISTRIBUTED ALGORITHMS Two distinctly different distributed algorithms are described in this section The first utilizes token passing to distribute the computation, and has a strong approximation ratio when applied to arbitrary graphs The second is a fully distributed algorithm based on moving away from simply computing... radio networks, Journal of High Speed Networks, 2: 405–423, 1993 18 X Ma, Broadcast Scheduling in Multi-hop Packet Radio Networks 2000 PhD Dissertation, University of Delaware 19 X Ma and E Lloyd, An incremental algorithm for broadcast scheduling in packet radio networks, in Proceedings IEEE MILCOM ‘98, 1998 20 X Ma and E L Lloyd, A distributed protocol for adaptive broadcast scheduling in packet radio... 165–167, 1990 4 J Flynn D Baker, A., Ephremedes The design and simulation of a mobile radio network with distributed control, IEEE Journal on Selected Areas in Communications, SAC-2: 226–237, 1999 5 A Ephremedis and T Truong, A distributed algorithm for efficient and interference free broadcasting in radio networks, In Proceedings IEEE INFOCOM, 1988 6 S Even, O Goldreich, S Moran, and P Tong, On the NP-completeness... Scheduling algorithms for multi-hop radio networks, IEEE/ ACM Transactions on Networking, 1: 166–177, 1993 370 BROADCAST SCHEDULING FOR TDMA IN WIRELESS MULTIHOP NETWORKS 25 S Ramanathan, Scheduling Algorithms for Multi-hop Radio Networks, 1992 PhD Dissertation, University of Delaware 26 R Ramaswami and K K Parhi, Distributed scheduling of broadcasts in a radio network, in Proceedings IEEE INFOCOM, 1989 27... a distributed fashion is straightforward: from the perspective of an individual station, its actions in regard to scheduling precisely follow the method outlined above The only complications arise in the communication aspects, including: station registration when joining the network, coordination of geographically close stations attempting to run the algorithm in an overlapping fashion, and coordination... given in [27] Unit disk graphs are a special type of intersection graph in which each circle has the same radius ț A 2-approximation algorithm is given in [14] for (r, s)-civilized graphs [29] Note that (r, s)-civilized graphs include intersection (hence, unit disk) graphs provided that there is a fixed minimum distance between the nodes For these special types of intersection and unit disk graphs, the... 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Lemma 4 For unit disk graphs, an optimal schedule uses at least 1 + D/13 colors Proof: Recall that D is the distance-2 degree of the graph Given a node v in a unit disk graph, all of its distance-2 neighbors lie within the interference region of v Partition the interference region of v into 13 sections of equal angle around v As in the proof of Theorem 7, each section satisfies the conditions of Lemma . by Algorithm_IR is much superior for unit disk graphs. 16.5 DISTRIBUTED ALGORITHMS Two distinctly different distributed algorithms are described in this. method, the nodes are placed into descending order according to their degrees. The nodes are then greedi- ly colored according to that ordering. The running